Sycamore Community School District was above the state’s average in multiple categories, including mathematics and English, according to the 2023 Illinois State Board of Education report card released on Monday.

Illinois Report Card: Sycamore students above state average in math, English language arts proficiency

Cambridge’s 11 candidates for the School Committee discussed their views on special education, achievement gaps, and the mathematics curriculum during various forums leading up to the Nov. 7 city municipal elections.

Cambridge School Committee Candidates Discuss Special Ed, Achievement Gaps, Math in Lead Up to Election

Join EnCorps’ STEMx Tutors program to virtually tutor math to middle school students who are under-performing in Foundational Math, Pre-Algebra and Algebra 1 ... live in the US' About EnCorps: EnCorps Inc., a Los Angeles-based nonprofit with operations ...

VIRTUAL: Volunteer Math Tutor

Educators are continually seeking innovative AI approaches to improve instruction, especially in fields like math and computer science. As Jake Price, an assistant professor of mathematics and computer science at the University of Puget Sound,

AI’s Growing Role in Classroom Problem-Solving

As the first step in a new partnership focused on equity in math education, Casio and Open Up Resources (OUR) have joined forces to integrate Casio's ClassPad.net technology into the Algebra 1 Program of Open Up High School Mathematics,

CASIO AND OPEN UP RESOURCES LAUNCH NEW INNOVATIVE PARTNERSHIP TO ENHANCE MATHEMATICS EDUCATION

Integration of Casio Technology Into Open Up's Algebra 1 Program Will Streamline Math Education Program; Improve Equity By Expanding Access to Less

Chegg is an education technology company that offers student services such as textbook rental and study help that's come far from its humble beginnings. But understanding all Chegg has to offer requires a bit more explanation.

What is Chegg?

For as long as Jake Price has been a teacher, Wolfram Alpha — a website that solves algebraic problems online — has threatened to make algebra homework obsolete. Teachers learned to work around and with it, said Price, assistant professor of ...

Chatbots might disrupt math and computer science classes. Some teachers see upsides

Professor Price doesn’t see ChatGPT as a threat, and he’s not alone. Some math professors believe AI, when used correctly, could help strengthen math instruction. And it’s arriving on the scene at a time when math scores are at a national historic low and educators are questioning if math should be taught differently.

AI may disrupt math and computer science classes. Is there an upside?

Racial achievement gaps in math worsened in the wake of the COVID-19 pandemic. Some schools are trying to level the playing field by ditching classes that separate students by achievement level.

Racial gaps in math have grown. A school tried closing theirs by teaching all kids the same classes

Algebra, Inc.

algebrainc

Everything You Need to Know About Zero-Knowledge Proof (ZKP) Technology

Privacy in Cloud Computing through Immersion-based Coding: Conclusion and References

Authors:
(1) Haleh Hayati, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands;
(2) Nathan van de Wouw, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands;
(3) Carlos Murguia, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands, and with the School of Electrical Engineering and Robotics, Queensland University of Technology, Brisbane, Australia. Table of Links Abstract and Introduction Problem Formulation Affine Solution to Problem 1 Privacy Guarantee General Guidelines for Implementation Case Studies Conclusion and References II. PROBLEM FORMULATION The setting that we consider is the following. The user aims to run an algorithm remotely (in the cloud) using her/his local data yk to produce some data utility uk (what the algorithm returns), see Fig. 1(a). However, the user does not want to share the original data yk as it contains sensitive private information that could be inferred. The goal of our proposed privacy-preserving scheme is to devise an encoding mechanism to mask the original data and the algorithm to be run in the cloud so that inference of sensitive information from the encoded disclosed data is as hard as possible (in a probabilistic Differential-Privacy (DP) sense) without degrading the performance of the encoded algorithm (so a homomorphic probabilistic scheme). A. Immersion-based Coding We now have all the ingredients to pose the problem we seek to address. The flowchart of the original algorithm and the target algorithm are shown in Fig. 1. Remark 1 Note that solutions to Problem 1 characterize a class of encoding maps and target algorithms for which we can design homomorphic encryption schemes (a prescriptive design, i.e., for given (f, g, ζ0)) that operate over the reals. However, this class is infinite-dimensional (over a function space). It leads to an underdetermined algebraic problem with an infinite dimensional solution space. To address this aspect of the problem, we impose structure on the maps we seek to design. We restrict to random affine maps composed of linear coordinate transformations and additive random processes. In what follows, we prove that this class of maps is sufficient to guarantee an arbitrary level of differential privacy. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Haleh Hayati, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (2) Nathan van de Wouw, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (3) Carlos Murguia, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands, and with the School of Electrical Engineering and Robotics, Queensland University of Technology, Brisbane, Australia. Authors: Authors: (1) Haleh Hayati, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (2) Nathan van de Wouw, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (3) Carlos Murguia, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands, and with the School of Electrical Engineering and Robotics, Queensland University of Technology, Brisbane, Australia. Table of Links Abstract and Introduction Abstract and Introduction Problem Formulation Problem Formulation Affine Solution to Problem 1 Affine Solution to Problem 1 Privacy Guarantee Privacy Guarantee General Guidelines for Implementation General Guidelines for Implementation Case Studies Case Studies Conclusion and References Conclusion and References II. PROBLEM FORMULATION The setting that we consider is the following. The user aims to run an algorithm remotely (in the cloud) using her/his local data yk to produce some data utility uk (what the algorithm returns), see Fig. 1(a). However, the user does not want to share the original data yk as it contains sensitive private information that could be inferred. The goal of our proposed privacy-preserving scheme is to devise an encoding mechanism to mask the original data and the algorithm to be run in the cloud so that inference of sensitive information from the encoded disclosed data is as hard as possible (in a probabilistic Differential-Privacy (DP) sense) without degrading the performance of the encoded algorithm (so a homomorphic probabilistic scheme). A. Immersion-based Coding A. Immersion-based Coding We now have all the ingredients to pose the problem we seek to address. The flowchart of the original algorithm and the target algorithm are shown in Fig. 1. Remark 1 Note that solutions to Problem 1 characterize a class of encoding maps and target algorithms for which we can design homomorphic encryption schemes (a prescriptive design, i.e., for given (f, g, ζ0)) that operate over the reals. However, this class is infinite-dimensional (over a function space). It leads to an underdetermined algebraic problem with an infinite dimensional solution space. To address this aspect of the problem, we impose structure on the maps we seek to design. We restrict to random affine maps composed of linear coordinate transformations and additive random processes. In what follows, we prove that this class of maps is sufficient to guarantee an arbitrary level of differential privacy. Remark 1 Note that solutions to Problem 1 characterize a class of encoding maps and target algorithms for which we can design homomorphic encryption schemes (a prescriptive design, i.e., for given (f, g, ζ0)) that operate over the reals. However, this class is infinite-dimensional (over a function space). It leads to an underdetermined algebraic problem with an infinite dimensional solution space. To address this aspect of the problem, we impose structure on the maps we seek to design. We restrict to random affine maps composed of linear coordinate transformations and additive random processes. In what follows, we prove that this class of maps is sufficient to guarantee an arbitrary level of differential privacy. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Privacy in Cloud Computing through Immersion-based Coding: Problem Formulation

Optimizing Prompts with LLMs: Key Findings and Future Directions

Table of Links Abstract and 1. Introduction Why ELIZA?
The Intelligence Engineers
Newell, Shaw, and Simon’s IPL Logic Theorist: The First True AIs
From IPL to SLIP and Lisp
A Critical Tangent into Gomoku
Interpretation is the Core of Intelligence
The Threads Come Together: Interpretation, Language, Lists, Graphs, and Recursion
Finally ELIZA: A Platform, Not a Chat Bot!
A Perfect Irony: A Lisp ELIZA Escapes and is Misinterpreted by the AI Community
Another Wave: A BASIC ELIZA turns the PC Generation on to AI
Conclusion: A certain danger lurks there
Acknowledgements and References 5 From IPL to SLIP and Lisp Recall (from McCorduck) that Weizenbaum was connected to AI through a number of paths, including Kenneth Colby, a Stanford psychiatrist who was interested in modeling neurosis and paranoia[22], and Ed Feigenbaum, a computer scientist at Berkeley. Feigenbaum had been a student of Simon’s, and was creating various AI programs in IPL.[26] Once Weizenbaum got to MIT, he became associated with Project MAC, which was begun, in part, by John McCarthy, the inventor of Lisp and the person who coined the term “Artificial Intelligence”[6].[3] But Weizenbaum was, first and foremost, what we would now call a software engineer[4], having just come from GE where he worked on highly practical programs. There were several projects mounted to build on the clear successes of Newell and Simon’s IPL work without having to cope with its ugliness and inefficiency. As mentioned above, there were already several much more programmer-friendly languages, notably Fortran and COBOL. But these languages were aimed at science, engineering, and business, and did not provide the AI-related functionalities of IPL, such as symbol processing, lists, and recursion. So the question naturally arose as to how to add these capabilities to those already-existing languages The inventive entanglement between Fortran, IPL, and Lisp is concisely captured in a brief mention by Gelernter and coworkers in creating FLPL, the “Fortran List Processing Language”: “[...] consideration was given to the translation of a JOHNNIAC IPL for use with the IBM 704 computer. However, J. McCarthy, who was then consulting for the project, suggested that Fortran could be adapted to serve the same purpose. He pointed out that the nesting of functions that is allowed within the Fortran format makes possible the construction of elaborate information-processing subroutines with a single statement. The authors have since discovered [...] the close analogy that exists between the structure of [a Newell, Shaw, and Simon] list and a certain class of algebraic expressions that may be written within the language. [...] Not to be overlooked is the considerable sophistication incorporated into the Fortran compiler itself, all of which carries over, of course, into our Fortran-compiled list-processing language. It is reasonable to estimate that a routine written in our language would run about five times as fast as the same program written in an interpretive language [like IPL].”[27, pp. 88–89] Like Gelernter, Weizenbaum implemented a set of IPL-inspired list-processing facilities as a set of Fortran-(and later MAD-)-callable functions, which he called SLIP. In creating SLIP, Weizenbaum took a route similar to Gelernter’s. In his 1963 paper, Weizenbaum (in addition to citing IPL and FLPL as influences) critiques McCarthy’s Lisp (although not mentioning it by name): “List processing has won a number of dedicated converts. Some have, however, become somewhat too fervent in their advocacy of list processing. While there may be some programming tasks which are best solved entirely within some list processing system, most tasks coming to the ordinary programmer require the application of a number of distinct techniques. The packaging of a variety of tools within a single tool box appears to be a good, if not an optimum, way of outfitting a worker setting out to solve complex problems. FORTRAN, ALGOL, and other languages of the same type provide excellent vehicles for such provisioning. Apart from the fact that they are very powerful in themselves, they have the advantage that they are well known. The task of coming to grips with these new techniques is then that of adding to a vocabulary of an already assimilated language rather than that of learning an entirely new one.”[49, pp. 535–536] The acknowledgement in the 1963 SLIP paper is worth quoting in full, as it makes many of the connections explicit: “[...] SLIP owes a considerable debt to previous list processing systems. Certain of its features are, however, more the result of attempts to build a symbol manipulator for the use of behavioral scientists than to generalize other processors. In this connection, the continuing and generous advice and support of Kenneth Colby, M.D., of Stanford University and of Dr. Edward Feigenbaum of the University of California at Berkeley is gratefully acknowledged. The author also wishes to thank Howard Sturgis of the University of California at Berkeley and Larry Breed of Stanford University for their parts in making the system operative on the computers at their respective computation centers.”[49, p. 536][5] Although the SLIP paper makes no mention of ELIZA, nor of any specific application, this acknowledgement makes clear that SLIP was motivated by “previous list processing systems” (specifically IPL-V, and FLPL), as a “symbol manipulator for the use of behavioral scientists”, and that SLIP was running on computers at both Stanford and Berkeley. Although the SLIP paper explicitly cites Gelernter’s FLPL, it is likely, given the above acknowledgement, close timing, and length of publication cycles, that SLIP was developed essentially simultaneously with FLPL, and that the citation was a publication-sequence nod, rather than FLPL having directly influenced SLIP’s development. Whereas Weizenbaum originally embedded SLIP for Fortran, the SLIP that ELIZA is written in was embedded in MAD (although using the same underlying foreignfunction calling machinery as Fortran – an intermediate language called “FAP” – the Fortran Assembly Program). Intriguingly, this SLIP implementation for the 7090 MAD programming language came from Yale[42, p. 62L1], suggesting some interaction with Gelernter, although the details are obscure. Notable from the above is McCarthy’s involvement with FLPL, and especially the point made by McCarthy about functional nesting of Fortran statements. McCarthy’s own approach to creating a high-level AI language was very different from the one he recommended to Gelernter, which was also adopted by Weizenbaum. In addition to his interest in formal matters of logic and mathematics, McCarthy was coincidentally playing around with Church’s lambda calculus, which, as discussed above, was the motivation for Turing’s work that led to universal computers. This aligned nicely with the functional nesting pointed out by McCarthy to Gelernter. Recursion is central to both universal computation and AI.[32] McCarthy, along with his students, most notably, Steve Russell, figured out, perhaps coincidentally, how to bring list processing and recursion together into an elegant programming paradigm that was far simpler and more elegant than IPL, FLPL or SLIP; thus, the birth of Lisp, among the world’s most influential and enduring programming languages.[19] Author:
(1) Jeff Shrager, Blue Dot Change and Stanford University Symbolic Systems Program (Adjunct)( jshrager@stanford.edu). This paper is available on arxiv under CC BY 4.0 license. [3] Although McCarthy moved to Stanford in 1962, two years before Weizenbaum arrived at MIT.[7] [4] That term was coined in the late 1960s.[18] [5] Symbols in Lisp and IPL-V are named pointers and those languages, being self-contained, have specific support for symbol manipulation. Because SLIP is embedded in another language (initially Fortran, later MAD in which ELIZA was implemented), the naming, should the programmer choose to do so, results from Fortran assignment to variables. It would require another whole paper to track the meaning of “symbol” through the history of programming languages, AI, and cognitive science. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Why ELIZA? The Intelligence Engineers Newell, Shaw, and Simon’s IPL Logic Theorist: The First True AIs From IPL to SLIP and Lisp A Critical Tangent into Gomoku Interpretation is the Core of Intelligence The Threads Come Together: Interpretation, Language, Lists, Graphs, and Recursion Finally ELIZA: A Platform, Not a Chat Bot! A Perfect Irony: A Lisp ELIZA Escapes and is Misinterpreted by the AI Community Another Wave: A BASIC ELIZA turns the PC Generation on to AI Conclusion: A certain danger lurks there Acknowledgements and References Why ELIZA? Why ELIZA? The Intelligence Engineers The Intelligence Engineers Newell, Shaw, and Simon’s IPL Logic Theorist: The First True AIs Newell, Shaw, and Simon’s IPL Logic Theorist: The First True AIs From IPL to SLIP and Lisp From IPL to SLIP and Lisp A Critical Tangent into Gomoku A Critical Tangent into Gomoku Interpretation is the Core of Intelligence Interpretation is the Core of Intelligence The Threads Come Together: Interpretation, Language, Lists, Graphs, and Recursion The Threads Come Together: Interpretation, Language, Lists, Graphs, and Recursion Finally ELIZA: A Platform, Not a Chat Bot! Finally ELIZA: A Platform, Not a Chat Bot! A Perfect Irony: A Lisp ELIZA Escapes and is Misinterpreted by the AI Community A Perfect Irony: A Lisp ELIZA Escapes and is Misinterpreted by the AI Community Another Wave: A BASIC ELIZA turns the PC Generation on to AI Another Wave: A BASIC ELIZA turns the PC Generation on to AI Conclusion: A certain danger lurks there Conclusion: A certain danger lurks there Acknowledgements and References Acknowledgements and References 5 From IPL to SLIP and Lisp Recall (from McCorduck) that Weizenbaum was connected to AI through a number of paths, including Kenneth Colby, a Stanford psychiatrist who was interested in modeling neurosis and paranoia[22], and Ed Feigenbaum, a computer scientist at Berkeley. Feigenbaum had been a student of Simon’s, and was creating various AI programs in IPL.[26] Once Weizenbaum got to MIT, he became associated with Project MAC, which was begun, in part, by John McCarthy, the inventor of Lisp and the person who coined the term “Artificial Intelligence”[6].[3] But Weizenbaum was, first and foremost, what we would now call a software engineer[4], having just come from GE where he worked on highly practical programs. There were several projects mounted to build on the clear successes of Newell and Simon’s IPL work without having to cope with its ugliness and inefficiency. As mentioned above, there were already several much more programmer-friendly languages, notably Fortran and COBOL. But these languages were aimed at science, engineering, and business, and did not provide the AI-related functionalities of IPL, such as symbol processing, lists, and recursion. So the question naturally arose as to how to add these capabilities to those already-existing languages The inventive entanglement between Fortran, IPL, and Lisp is concisely captured in a brief mention by Gelernter and coworkers in creating FLPL, the “Fortran List Processing Language”: “[...] consideration was given to the translation of a JOHNNIAC IPL for use with the IBM 704 computer. However, J. McCarthy, who was then consulting for the project, suggested that Fortran could be adapted to serve the same purpose. He pointed out that the nesting of functions that is allowed within the Fortran format makes possible the construction of elaborate information-processing subroutines with a single statement. The authors have since discovered [...] the close analogy that exists between the structure of [a Newell, Shaw, and Simon] list and a certain class of algebraic expressions that may be written within the language. [...] Not to be overlooked is the considerable sophistication incorporated into the Fortran compiler itself, all of which carries over, of course, into our Fortran-compiled list-processing language. It is reasonable to estimate that a routine written in our language would run about five times as fast as the same program written in an interpretive language [like IPL].”[27, pp. 88–89] Like Gelernter, Weizenbaum implemented a set of IPL-inspired list-processing facilities as a set of Fortran-(and later MAD-)-callable functions, which he called SLIP. In creating SLIP, Weizenbaum took a route similar to Gelernter’s. In his 1963 paper, Weizenbaum (in addition to citing IPL and FLPL as influences) critiques McCarthy’s Lisp (although not mentioning it by name): “List processing has won a number of dedicated converts. Some have, however, become somewhat too fervent in their advocacy of list processing. While there may be some programming tasks which are best solved entirely within some list processing system, most tasks coming to the ordinary programmer require the application of a number of distinct techniques. The packaging of a variety of tools within a single tool box appears to be a good, if not an optimum, way of outfitting a worker setting out to solve complex problems. FORTRAN, ALGOL, and other languages of the same type provide excellent vehicles for such provisioning. Apart from the fact that they are very powerful in themselves, they have the advantage that they are well known. The task of coming to grips with these new techniques is then that of adding to a vocabulary of an already assimilated language rather than that of learning an entirely new one.”[49, pp. 535–536] The acknowledgement in the 1963 SLIP paper is worth quoting in full, as it makes many of the connections explicit: “[...] SLIP owes a considerable debt to previous list processing systems. Certain of its features are, however, more the result of attempts to build a symbol manipulator for the use of behavioral scientists than to generalize other processors. In this connection, the continuing and generous advice and support of Kenneth Colby, M.D., of Stanford University and of Dr. Edward Feigenbaum of the University of California at Berkeley is gratefully acknowledged. The author also wishes to thank Howard Sturgis of the University of California at Berkeley and Larry Breed of Stanford University for their parts in making the system operative on the computers at their respective computation centers.”[49, p. 536][5] Although the SLIP paper makes no mention of ELIZA, nor of any specific application, this acknowledgement makes clear that SLIP was motivated by “previous list processing systems” (specifically IPL-V, and FLPL), as a “symbol manipulator for the use of behavioral scientists”, and that SLIP was running on computers at both Stanford and Berkeley. Although the SLIP paper explicitly cites Gelernter’s FLPL, it is likely, given the above acknowledgement, close timing, and length of publication cycles, that SLIP was developed essentially simultaneously with FLPL, and that the citation was a publication-sequence nod, rather than FLPL having directly influenced SLIP’s development. Whereas Weizenbaum originally embedded SLIP for Fortran, the SLIP that ELIZA is written in was embedded in MAD (although using the same underlying foreignfunction calling machinery as Fortran – an intermediate language called “FAP” – the Fortran Assembly Program). Intriguingly, this SLIP implementation for the 7090 MAD programming language came from Yale[42, p. 62L1], suggesting some interaction with Gelernter, although the details are obscure. Notable from the above is McCarthy’s involvement with FLPL, and especially the point made by McCarthy about functional nesting of Fortran statements. McCarthy’s own approach to creating a high-level AI language was very different from the one he recommended to Gelernter, which was also adopted by Weizenbaum. In addition to his interest in formal matters of logic and mathematics, McCarthy was coincidentally playing around with Church’s lambda calculus, which, as discussed above, was the motivation for Turing’s work that led to universal computers. This aligned nicely with the functional nesting pointed out by McCarthy to Gelernter. Recursion is central to both universal computation and AI.[32] McCarthy, along with his students, most notably, Steve Russell, figured out, perhaps coincidentally, how to bring list processing and recursion together into an elegant programming paradigm that was far simpler and more elegant than IPL, FLPL or SLIP; thus, the birth of Lisp, among the world’s most influential and enduring programming languages.[19] Author: (1) Jeff Shrager, Blue Dot Change and Stanford University Symbolic Systems Program (Adjunct)( jshrager@stanford.edu). Author: Author: (1) Jeff Shrager, Blue Dot Change and Stanford University Symbolic Systems Program (Adjunct)( jshrager@stanford.edu). This paper is available on arxiv under CC BY 4.0 license. This paper is available on arxiv under CC BY 4.0 license. available on arxiv available on arxiv [3] Although McCarthy moved to Stanford in 1962, two years before Weizenbaum arrived at MIT.[7] [4] That term was coined in the late 1960s.[18] [5] Symbols in Lisp and IPL-V are named pointers and those languages, being self-contained, have specific support for symbol manipulation. Because SLIP is embedded in another language (initially Fortran, later MAD in which ELIZA was implemented), the naming, should the programmer choose to do so, results from Fortran assignment to variables. It would require another whole paper to track the meaning of “symbol” through the history of programming languages, AI, and cognitive science.

SLIP and Lisp: The Trailblazers of AI Programming and Symbolic Computation

A New Framework for Trustworthy AI Deductive Reasoning

Where does In-context Translation Happen in Large Language Models: Conclusion

Authors:
(1) Matthew Sprintson
(2) Edward Oughton Table of Links Abstract and Introduction 2. Literature Review 2.1 Reviewing Broadband Infrastructure’s Impact on the Economy 2.2 Previous Research into IO Modeling of Broadband Investment 2.3 Context of the Bipartisan Infrastructure Act through Previous Research 3. Methods and 3.1 Leontief Input-Output (IO) Modeling 3.2 Ghosh Supply-Side Assessment Methods for Infrastructure 3.3 Data and Application 4. Results and 4.1 To what extent does the Bipartisan Infrastructure Law allocate funding to unconnected communities in need? 4.2 What are the GDP impacts of the three funding programs within the Bipartisan Infrastructure Law? 4.3 How are the supply chain linkages affected by allocations from the Bipartisan Infrastructure Law? 5. Discussion 5.1 To what extent does the Bipartisan Infrastructure Law allocate funding to unconnected communities in need? 5.2 What are the GDP impacts of the three funding programs within the Bipartisan Infrastructure Law? 5.3 How are supply chain linkages affected by allocations from the Bipartisan Infrastructure Law?" Conclusion Acknowledgements and References 3 Methods In this section, various macroeconomic methods for assessing the contribution of infrastructure to the economy will be assessed. We begin by articulating a general macroeconomic system model based on IO, before detailing different demand-side and supply-side evaluation methods for broadband infrastructure. 3.1 Leontief Input-Output (IO) Modeling The IO approach begins by dividing the economy into n sectors, such as agriculture, manufacturing, energy, telecommunications, and other such industries. The model denotes inter-sectoral transfers as zij , representing the dollar value of transfers from sector i to sector j. For example, if sector 1 is electricity and sector 2 is telecommunications, z12 represents the dollar value of electricity sold to the telecommunications industry as an input. The other metric is fi , which denotes the final demand for the good or service from sector i. The final demand includes purchases by consumers, investments from businesses, purchases from the government, and net exports. The final output of sector i is denoted as xi , measured here in dollars, and represents the sum of all monetary values of the transactions from other inter-industry sectors, as well as the final demand for goods or services, stated in equation (1) as follows. Thus, we can list the linear equations of all n industries in rows, as detailed in equation (2). Moreover, the vector for the final output is x, as outlined in equation (3). Likewise, the matrix of inter-industry transactions and final demand can be represented as Z and F, respectively, as we present in equations (4) and (5). As each sector of the economy can be modeled in a row of the matrices above, the total expression for the total n industries in the economy can be modeled in a system of linear equations represented in equation (6). Where i represents an identity matrix of dimensions n by n. Each row of the matrix Z represents the aggregate output of its respective sector. To measure spillovers across industries, we can analyze technical coefficients aij = zij /xj . For two industries i and j, the technical coefficient aij is equal to the dollar's worth of input from i to the output of sector j. Because zij = aij * xj , we can replace the respective coefficients algebraically, as detailed in equation (7). Distributing, as per equation (8). Factoring, as per equation (9). Thus, as per the system of linear equations (10). Where A is the n by n matrix of technical coefficients as represented below, in equation (11). And the inverse of (I-A) is denoted as L, the Leontief-inverse matrix, as per equation (12 Therefore, we can use this formula to quantify the output generated by a shift in final demand for individual industrial sectors, or the whole economy. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Matthew Sprintson (2) Edward Oughton Authors: Authors: (1) Matthew Sprintson (2) Edward Oughton Table of Links Abstract and Introduction Abstract and Introduction 2. Literature Review 2.1 Reviewing Broadband Infrastructure’s Impact on the Economy 2.1 Reviewing Broadband Infrastructure’s Impact on the Economy 2.2 Previous Research into IO Modeling of Broadband Investment 2.2 Previous Research into IO Modeling of Broadband Investment 2.3 Context of the Bipartisan Infrastructure Act through Previous Research 2.3 Context of the Bipartisan Infrastructure Act through Previous Research 3. Methods and 3.1 Leontief Input-Output (IO) Modeling 3. Methods and 3.1 Leontief Input-Output (IO) Modeling 3.2 Ghosh Supply-Side Assessment Methods for Infrastructure 3.2 Ghosh Supply-Side Assessment Methods for Infrastructure 3.3 Data and Application 3.3 Data and Application 4. Results and 4.1 To what extent does the Bipartisan Infrastructure Law allocate funding to unconnected communities in need? 4. Results and 4.1 To what extent does the Bipartisan Infrastructure Law allocate funding to unconnected communities in need? 4.2 What are the GDP impacts of the three funding programs within the Bipartisan Infrastructure Law? 4.2 What are the GDP impacts of the three funding programs within the Bipartisan Infrastructure Law? 4.3 How are the supply chain linkages affected by allocations from the Bipartisan Infrastructure Law? 4.3 How are the supply chain linkages affected by allocations from the Bipartisan Infrastructure Law? 5. Discussion 5.1 To what extent does the Bipartisan Infrastructure Law allocate funding to unconnected communities in need? 5.1 To what extent does the Bipartisan Infrastructure Law allocate funding to unconnected communities in need? 5.2 What are the GDP impacts of the three funding programs within the Bipartisan Infrastructure Law? 5.2 What are the GDP impacts of the three funding programs within the Bipartisan Infrastructure Law? 5.3 How are supply chain linkages affected by allocations from the Bipartisan Infrastructure Law?" 5.3 How are supply chain linkages affected by allocations from the Bipartisan Infrastructure Law?" Conclusion Conclusion Acknowledgements and References Acknowledgements and References 3 Methods In this section, various macroeconomic methods for assessing the contribution of infrastructure to the economy will be assessed. We begin by articulating a general macroeconomic system model based on IO, before detailing different demand-side and supply-side evaluation methods for broadband infrastructure. 3.1 Leontief Input-Output (IO) Modeling The IO approach begins by dividing the economy into n sectors, such as agriculture, manufacturing, energy, telecommunications, and other such industries. The model denotes inter-sectoral transfers as zij , representing the dollar value of transfers from sector i to sector j. For example, if sector 1 is electricity and sector 2 is telecommunications, z12 represents the dollar value of electricity sold to the telecommunications industry as an input. The other metric is fi , which denotes the final demand for the good or service from sector i. The final demand includes purchases by consumers, investments from businesses, purchases from the government, and net exports. The final output of sector i is denoted as xi , measured here in dollars, and represents the sum of all monetary values of the transactions from other inter-industry sectors, as well as the final demand for goods or services, stated in equation (1) as follows. Thus, we can list the linear equations of all n industries in rows, as detailed in equation (2). Moreover, the vector for the final output is x, as outlined in equation (3). Likewise, the matrix of inter-industry transactions and final demand can be represented as Z and F, respectively, as we present in equations (4) and (5). As each sector of the economy can be modeled in a row of the matrices above, the total expression for the total n industries in the economy can be modeled in a system of linear equations represented in equation (6). Where i represents an identity matrix of dimensions n by n. Each row of the matrix Z represents the aggregate output of its respective sector. To measure spillovers across industries, we can analyze technical coefficients aij = zij /xj . For two industries i and j, the technical coefficient aij is equal to the dollar's worth of input from i to the output of sector j. Because zij = aij * xj , we can replace the respective coefficients algebraically, as detailed in equation (7). Distributing, as per equation (8). Factoring, as per equation (9). Thus, as per the system of linear equations (10). Where A is the n by n matrix of technical coefficients as represented below, in equation (11). And the inverse of (I-A) is denoted as L, the Leontief-inverse matrix, as per equation (12 Therefore, we can use this formula to quantify the output generated by a shift in final demand for individual industrial sectors, or the whole economy. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Understanding Leontief Input-Output Modeling in Infrastructure Economics

Author:
(1) Attila Losonczi. Table of Links Abstract and 1 Introduction 1.1 Basic notions and notations 1.2 Basic definitions from [7] and [8] 2 Generalized integral 2.1 Multiplication on [0, +∞) × [−∞, +∞] 2.2 Measurability 2.3 The integral of functions taking values in [0, +∞) × [0, +∞) 3 Applications 4 References 2.1 Multiplication on [0, +∞) × [−∞, +∞] In order to define the integral, we will need multiplication on [0, +∞) × [0, +∞). We will use a natural way to do that, however one might argue against it, saying that that rule does not always hold for Hausdorff dimension and measure. We admit that it is true, but we could not find a better way until now, and more importantly, this method of multiplication and the resulting integral suit our later purposes completely. We define the operation on a greater domain that we will actually need. Remark 2.2. The multiplication of a real number and a value from [0, +∞) × [−∞, +∞] is a special case of the multiplication defined here, because for c ∈ R,(d, m) ∈ [0, +∞)×[−∞, +∞], we have c(d, m) = (0, c)(d, m) = (d, cm) if c 6= 0, while if c = 0, then both sides equal to (0, 0). Therefore if we identify c with (0, c), then we can embed the former algebraic structure into the current one. Proof. The sufficiency is obvious. Proposition 2.4. The multiplication is commutative and associative. Proof. Being commutative is trivial. Proposition 2.5. The distributive law holds in [0, +∞) × [0, +∞] Example 2.6. The distributive law does not hold in general if we allow negative numbers in the second coordinate as well. See e.g. (1, 1) (0, 5) + (0, −5)) = (0, 0) 6= (1, 5) + (1, −5) = (1, 0). This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Author: (1) Attila Losonczi. Author: Author: (1) Attila Losonczi. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 Basic notions and notations 1.1 Basic notions and notations 1.2 Basic definitions from [7] and [8] 1.2 Basic definitions from [7] and [8] 2 Generalized integral 2.1 Multiplication on [0, +∞) × [−∞, +∞] 2.1 Multiplication on [0, +∞) × [−∞, +∞] 2.2 Measurability 2.2 Measurability 2.3 The integral of functions taking values in [0, +∞) × [0, +∞) 2.3 The integral of functions taking values in [0, +∞) × [0, +∞) 3 Applications 3 Applications 4 References 4 References 2.1 Multiplication on [0, +∞) × [−∞, +∞] In order to define the integral, we will need multiplication on [0, +∞) × [0, +∞). We will use a natural way to do that, however one might argue against it, saying that that rule does not always hold for Hausdorff dimension and measure. We admit that it is true, but we could not find a better way until now, and more importantly, this method of multiplication and the resulting integral suit our later purposes completely. We define the operation on a greater domain that we will actually need. Remark 2.2. The multiplication of a real number and a value from [0, +∞) × [−∞, +∞] is a special case of the multiplication defined here, because for c ∈ R,(d, m) ∈ [0, +∞)×[−∞, +∞], we have c(d, m) = (0, c)(d, m) = (d, cm) if c 6= 0, while if c = 0, then both sides equal to (0, 0). Therefore if we identify c with (0, c), then we can embed the former algebraic structure into the current one. Remark 2.2. Proof . The sufficiency is obvious. Proof Proposition 2.4. The multiplication is commutative and associative. Proposition 2.4. The multiplication is commutative and associative. Proof . Being commutative is trivial. Proof Proposition 2.5. The distributive law holds in [0, +∞) × [0, +∞] Proposition 2.5. The distributive law holds in [0, +∞) × [0, +∞] Example 2.6. The distributive law does not hold in general if we allow negative numbers in the second coordinate as well. Example 2.6. The distributive law does not hold in general if we allow negative numbers in the second coordinate as well. See e.g. (1, 1) (0, 5) + (0, −5)) = (0, 0) 6= (1, 5) + (1, −5) = (1, 0). See e.g. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. available on arxiv

Stories, News & Mentions about ALGEBRAINC

Algebrainc’s Stories on HackerNoon

Algebrainc's News & Mentions